The determinant is the (essentially unique) universal alternating multilinear map.
be the 1-dimensional sign representation? on the symmetric group , taking each transposition to . We may linearly extend the sign action of , so that names a (right) -module with underlying vector space . At the same time, acts on the tensor product of a vector space by permuting tensor factors, giving a left -module structure on . We define the Schur functor
by the formula
It is called the alternating power (of ).
whose values are naturally regarded as graded by degree .
This applies in particular to the category of supervector spaces; if is a supervector space concentrated in odd degree, say with component , then each symmetry maps for elements . It follows that the graded component is concentrated in degree, with component .
There is a canonical natural isomorphism .
Again take to be the category of supervector spaces. Since the left adjoint preserves coproducts and since the tensor product of provides the coproduct for commutative monoid objects, we have a natural isomorphism
Examining the grade component , this leads to an identification
and now the result follows by considering the case where are concentrated in odd degree.
If is -dimensional, then has dimension . In particular, is 1-dimensional.
By induction on dimension. If , we have that and are -dimensional, and clearly for , at least when .
We then infer
where the dimensions satisfy the same recurrence relation as for binomial coefficients: .
(consider the transposition in which swaps and ) and so we may take only such expressions on the left where as forming a spanning set for , and indeed these form a basis. The number of such expressions is .
In the case where , the same development may be carried out by simply decreeing that whenever for some pair of distinct indices , .
Now let be an -dimensional space, and let be a linear map. By the proposition, the map
We see then that if is of dimension ,
that takes products of matrices to products in . The determinant however is of course independent of choice of basis, since any two choices are related by a change-of-basis matrix , where and its transform have the same determinant.
By following the definitions above, we can give an explicit formula:
We work over fields of arbitrary characteristic. The determinant satisfies the following properties, which taken together uniquely characterize the determinant. Write a square matrix as a row of column vectors .
is separately linear in each column vector:
whenever for distinct .
, where is the identity matrix.
Other properties may be worked out, starting from the explicit formula or otherwise:
If is a diagonal matrix, then is the product of its diagonal entries.
More generally, if is an upper (or lower) triangular matrix, then is the product of the diagonal entries.
If is a field extension and is a -linear map , then . Using the preceding properties and the Jordan normal form? of a matrix, this means that is the product of its eigenvalues (counted with multiplicity), as computed in the algebraic closure of .
If is the transpose of , then .
A simple observation which flows from these basic properties is
Let be column vectors of dimension , and suppose
Then for each we have
where occurs as the column vector on the right.
This follows straightforwardly from properties 1 and 2 above.
For instance, given a square matrix such that , and writing , this allows us to solve for a vector in an equation
and we easily conclude that is invertible if .
This holds true even if we replace the field by an arbitrary commutative ring , and we replace the condition by the condition that is a unit. (The entire development given above goes through, mutatis mutandis.)
Given a linear endomorphism of a finite rank free unital module over a commutative unital ring, one can consider the zeros of the characteristic polynomial . The coefficients of the polynomial are the concern of the Cayley-Hamilton theorem.
A useful intuition to have for determinants of real matrices is that they measure change of volume. That is, an matrix with real entries will map a standard unit cube in to a parallelpiped in (quashed to lie in a hyperplane if the matrix is singular), and the determinant is, up to sign, the volume of the parallelpiped. It is easy to convince oneself of this in the planar case by a simple dissection of a parallelogram, rearranging the dissected pieces in the style of Euclid to form a rectangle. In algebraic terms, the dissection and rearrangement amount to applying shearing or elementary column operations to the matrix which, by the properties discussed earlier, leave the determinant unchanged. These operations transform the matrix into a diagonal matrix whose determinant is the area of the corresponding rectangle. This procedure easily generalizes to dimensions.
The sign itself is a matter of interest. An invertible transformation is said to be orientation-preserving if is positive, and orientation-reversing if is negative. Orientations play an important role throughout geometry and algebraic topology, for example in the study of orientable manifolds (where the tangent bundle as -bundle can be lifted to a -bundle structure, being the subgroup of matrices of positive determinant). See also KO-theory.
Finally, we include one more property of determinants which pertains to matrices with real coefficients (which works slightly more generally for matrices with coefficients in a local field):
see Pfaffian for the moment